Analysis of Tooth Error of Hardened Gear Tooth Machining
1 The theoretical calculation formula of tooth profile error
When the hard tooth surface involute cylindrical gears are geared, the main source of the tooth profile error can be divided into the error caused by the gear tool and the error caused by the processing machine tool, that is, the tooth form error can be expressed as
Dff=[(Df tool)2+(Df machine tool)2]1â„2
Among them, the error caused by the gear tool Df tool also includes the tool original tooth profile error Dfa0, the rake face non-radial error Dfg, the hob axial pitch error Dft knife, the blade spiral error DfS knife, the tool installation error Df Tool installation, etc.; Errors caused by machining machines Df Machine tools also include machine motion error Df movement, machine tool elastic deformation error Df deformation, workpiece mounting error Df workpiece mounting. Therefore, one of the main measures to improve the precision of the tooth profile of the machined gear is to increase the precision of the tool manufacturing; the second is to improve the machine tool movement accuracy (especially the worktable rotation accuracy).
Fig. 1 The original tooth angle with front face of 0°
Figure 2 The original tooth angle of -30° on the front blade surface
Figure 3 blade shape
Figure 4 Workpiece Tooth Error
Fig. 5 Torsional vibration error calculation graph of machine table
2 Tooth shape error due to tool manufacturing error
The structure of a hard alloy roller razor used for machining hardened tooth surfaces is different from that of a shovel-type standard hob. The hardened surface roller razor has a front blade surface of -30° (a standard hob is 0°) and a number of teeth of about 30 to 40 (a standard hob is about 10). Therefore, the side edge of the hard tooth surface roller razor cannot be ground. The diamond grinding wheel is usually ground by spiral surface grinding to ensure that each blade is distributed on the basic worm surface of the tool. In many tool manufacturing error projects, the original tooth angle error Dfa0 and the knife edge spiral error DfS knife are the most difficult to control and are analyzed as follows:
1. Original tooth angle error Dfa0
The original tooth angle a0 is calculated accurately based on the tooling principle. For a hob whose front blade surface is 0°, the original tooth angle is the two fixed values ​​a0 left and a0 right (see Figure 1).
The front blade surface of the hardened roller razor is -30°, and the blade shape formed by its intersection with the basic worm surface is a curve, and the original tooth angles of each point on the curve are different (see FIG. 2 ).
In the manufacture of hard-tooth roller razors, the blades are first pre-ground into a straight shape, while the back is opened (see Figure 3a, d) and then precision ground (see Figures 3b, c). After the formation of the blade edge from f to f', wider than the original several times. Due to the uneven grinding margin, the finely ground blade will generate a tooth profile error, resulting in overcutting of the toothed center of the blade. Therefore, the actual tooth shape of the workpiece processed by this hard tooth surface rolling razor is mostly raised (see FIG. 4).
2. Blade spiral error DfS knife
Hard tooth roller razors are welded constructions. After the welding, the position of each blade must be changed, resulting in uneven machining allowance during finish grinding and it is difficult to ensure the required machining accuracy. In addition, the poor linearity (about 0.015mm) of the diamond grinding wheel used for precision grinding roller razors is also one of the important causes of tool manufacturing errors.
In summary, the tooth profile error caused by the tool manufacturing error (measurement error value of about 0.015 to 0.020 mm) is: Df tool = 0.015 to 0.020 mm.
3 Tooth shape error due to machine tool motion error
In hard tooth gear machining, the movement error of the processing machine has a great influence on the tooth profile error of the workpiece. Through inspection of the tested workpieces, it was found that the tooth shape error of the hard tooth face gear machining fluctuates compared with the ordinary gear processing, indicating that the hard tooth surface hobbing precision is very sensitive to the gear hardness, and the dynamic characteristics of the processing system are Tooth error has a decisive influence. In the motion error of the machine tool, the most significant influence on the tooth profile error is the torsional vibration error of the machine tool table and the displacement error of the front and rear of the machine tool in the radial direction of the gear.
1. Tooth shape error due to torsional vibration of the machine table
The torsional vibration of the gear processing machine table will directly affect the tooth profile error of the machined gear. This error is detected in the tangential direction of the gear. The calculation figure is shown in Figure 5. In the drawing, the workpiece is the diameter of the machined gear, Da< is the torsional vibration angle of the workpiece when the machine tool is cutting, and Da is the torsional vibration angle of the workpiece when the machine tool is not being cut.
From Fig. 5, it can be seen that the transmission error when the machine tool is not cutting is DS1'=r workpiece tanDa'=(176.25/2)tan0.0083°=0.013mm; the movement error when the machine tool is cutting is DS1=r workpiece tanDa=(176.25/ 2) tan0.021°=0.032mm. In the formula, Da'=0.0083° and Da=0.021° are measured experimentally. Since DS1 2. Tooth shape error caused by the error in the radial direction of the gears in front and rear of the machine tool
Through the test, the displacement error of the front and rear columns of the machine tool in the radial direction of the gear is DS2=0.015mm (see Figure 6). The resulting tooth profile error Dfs=DS2cos20°=0.014mm.
Therefore, the tooth profile error caused by the machine motion error is
Df machine tool = [(Dfs)2+(DS1)2]1â„2=[(0.014)2+(0.032)2]1â„2=0.035mm
The tooth profile error caused by tool manufacturing error and machine motion error is
Dff=[(Df tool)2+(Df machine tool)2]1â„2=(0.0202+0.0352)1â„2=0.041mm
Fig. 6 Deformation of the front and rear pillars of the machine tool in the radial direction of the gear
Figure 7 blade positioning slot structure
Figure 8 The tooth profile
Figure 9 Torsion angle of gear mounting shaft under cutting force
4 Measures to Reduce Hardened Tooth Profile Errors
1. Reduce tool manufacturing errors
In order to reduce the manufacturing error of gear cutting caused by gear cutting tools, it is necessary to improve the manufacturing precision of the hard tooth rolling razor. To this end, the following measures can be taken: 1 To improve the precision of the preform of the blade blank; 2 To change the blade to a positioning groove Structure (see Figure 7); 3 Calculate the knife edge curve, and open the front and back knife faces of the knife edge according to the approximate curve.
2. Reduce machine movement error
1. Hard-tooth hobbing process judgment In the hard-tooth hobbing process, if the machine tool movement error is too large, it will seriously affect the normal cutting. Generally, it can be judged whether the cutting is normal or not by observing the tooth profile. If the knife pattern is neat and continuous (see Fig. 8a), it indicates that the cutting is normal; if the knife pattern is hidden, there are traces of squeezing (see Fig. 8b), which indicates that some of the cutting edges are not normal; if the tooth surface is disordered or not visible at all Knife marks indicate that the cutting is abnormal.
2. Calculation of Machine Motion System Error
The dynamic error in the hard-tooth hobbing process is mainly manifested in the torsional vibration of the machine tool and the corresponding deformation of the support member. When cutting gears with carbide tools, the range of feed of machining teeth is usually f = 0.1-0.2mm/r, and the corresponding cutting force range is 15-22kg. For hardened gears with hardness greater than 60HRC, the cutting force corresponding to f = 0.1mm/r is 150kg; the cutting force corresponding to f = 0.2mm/r can reach 200kg. Generally, the hard-tooth roll cutting feed rate is f=0.2-0.4mm/r, so the amount of torsional vibration of the computer bed and the deformation of the supporting member caused by the cutting force of 200kg can be used.
1. Torsional vibration due to cutting force
As shown in Fig.9, the gear mounting shaft will produce a torsion angle Y under the action of cutting force F (F=200kg). Its calculation formula is
Ymax = (Mnmax/GIp) (180°/p)
Where: Mnmax - the maximum torque applied to the gear mounting shaft, Mnmax = Fd/2 = 1.725 × 105 N · mm
G——Shear elastic modulus, taking G=79.4 GPa
Ip - the moment of inertia of the gear mounting shaft, Ip=pd04/32=1.27×106mm4
d - gear diameter, d = 176.25mm
Figure 10 Force model of gear mounting shaft
D0——diameter of gear mounting shaft, d0=60mm
With the substitution parameters, Ymax=0.98°/m can be obtained, and the twist angle Y′max=0.15° at 150 mm of the gear mounting shaft. From this, it is possible to calculate the amount of torsional vibration of the gear mounting shaft caused by the cutting force DS=d/2tanY'max=176.25/2 tan 0.15°=0.23 mm. From the calculation results, it can be seen that under the action of the cutting force F, the torsion angle (Y'max=0.15°) of the gear mounting axis (Ø60 mm) is large, which causes a large error in the machine motion system. In order to solve this problem, a Ø120mm sleeve can be installed on the gear mounting shaft. The torsion angle Y'max=0.0098° obtained by this calculation can greatly reduce the error of the machine motion system.
2. Bending deformation of gear mounting shaft caused by cutting force
The cutting force F is converted into a force P acting in the radial direction of the gear, P=F/tanβ, and the tooth angle β=20°, then P=200/tan20°=550 kg. When calculating the radial displacement of the gear under the action of P=550kg, the stress condition of the gear mounting shaft is simplified to the model shown in Fig. 10. Based on this, the bending deformation of the gear mounting shaft can be calculated as
Yc=Yc1-Yc2
Where: Yc1 - Deflection due to force P at point C, Yc1 = Pa3/3EI
Yc2——deflection caused by PB at C point, Yc2=PBa2(3L-a)/6EI
I - section moment of inertia of the shaft, I=pd4/64
E——elastic modulus of the shaft, E=206GPa
L - the total length of the shaft, L = 660mm
A—distance of cutting force point C to fixed end A, a=380mm
D——The diameter of the shaft, D=60mm
PB——Support force at point B, PB=Pa2(3L-a)/2L3=221kg
1
Figure 11 Rear column force model
By substituting each parameter value into the formula, the bending distortion of the gear mounting shaft caused by the cutting force can be calculated as Yc=0.117mm.
3. Post-bending deformation caused by cutting force
The bending deformation of the rear column caused by the cutting force can be calculated according to the force model shown in Figure 11.
YB=PBa3/3EI
In the formula: PB—B point reaction force, PB=221kg
a—The distance between the fixed end and the support hole, a=660mm
E—elastic modulus of the column, E=135 GPa
I—section moment of the column, I=( BH3-bh3)/12=9.7x108mm4
B, H - column section size, B = 550mm, H = 335mm
t——The wall thickness of the column, t=32mm
b=B-2t=550-2×32=486mm,
h=H-2t=335-2×32=271mm
Substituting each parameter value into the formula, the post-pillar bending deformation amount YB=0.0016mm caused by the cutting force can be calculated.
From the above analysis and calculation, it can be seen that the gear mounting shaft and the column stiffness of the processing machine have a great influence on the processing accuracy of the gear. Therefore, in order to effectively reduce the tooth profile error of the gear being machined, it is necessary to increase the rigidity of the gear mounting shaft and the column, and to avoid or reduce the torsional vibration and the position change of the support member under the action of the cutting force as much as possible.
After the semi-finishing of the hardened gear tooth surface with a hard alloy roller razor, a fine grinding process is also required to ensure the gear machining accuracy. Through the precise installation and commissioning of the process system (machine tools—fixtures—tools—workpieces), the machining requirements for hardened tooth surfaces can be guaranteed.
When the hard tooth surface involute cylindrical gears are geared, the main source of the tooth profile error can be divided into the error caused by the gear tool and the error caused by the processing machine tool, that is, the tooth form error can be expressed as
Dff=[(Df tool)2+(Df machine tool)2]1â„2
Among them, the error caused by the gear tool Df tool also includes the tool original tooth profile error Dfa0, the rake face non-radial error Dfg, the hob axial pitch error Dft knife, the blade spiral error DfS knife, the tool installation error Df Tool installation, etc.; Errors caused by machining machines Df Machine tools also include machine motion error Df movement, machine tool elastic deformation error Df deformation, workpiece mounting error Df workpiece mounting. Therefore, one of the main measures to improve the precision of the tooth profile of the machined gear is to increase the precision of the tool manufacturing; the second is to improve the machine tool movement accuracy (especially the worktable rotation accuracy).
Fig. 1 The original tooth angle with front face of 0°
Figure 2 The original tooth angle of -30° on the front blade surface
Figure 3 blade shape
Figure 4 Workpiece Tooth Error
Fig. 5 Torsional vibration error calculation graph of machine table
2 Tooth shape error due to tool manufacturing error
The structure of a hard alloy roller razor used for machining hardened tooth surfaces is different from that of a shovel-type standard hob. The hardened surface roller razor has a front blade surface of -30° (a standard hob is 0°) and a number of teeth of about 30 to 40 (a standard hob is about 10). Therefore, the side edge of the hard tooth surface roller razor cannot be ground. The diamond grinding wheel is usually ground by spiral surface grinding to ensure that each blade is distributed on the basic worm surface of the tool. In many tool manufacturing error projects, the original tooth angle error Dfa0 and the knife edge spiral error DfS knife are the most difficult to control and are analyzed as follows:
1. Original tooth angle error Dfa0
The original tooth angle a0 is calculated accurately based on the tooling principle. For a hob whose front blade surface is 0°, the original tooth angle is the two fixed values ​​a0 left and a0 right (see Figure 1).
The front blade surface of the hardened roller razor is -30°, and the blade shape formed by its intersection with the basic worm surface is a curve, and the original tooth angles of each point on the curve are different (see FIG. 2 ).
In the manufacture of hard-tooth roller razors, the blades are first pre-ground into a straight shape, while the back is opened (see Figure 3a, d) and then precision ground (see Figures 3b, c). After the formation of the blade edge from f to f', wider than the original several times. Due to the uneven grinding margin, the finely ground blade will generate a tooth profile error, resulting in overcutting of the toothed center of the blade. Therefore, the actual tooth shape of the workpiece processed by this hard tooth surface rolling razor is mostly raised (see FIG. 4).
2. Blade spiral error DfS knife
Hard tooth roller razors are welded constructions. After the welding, the position of each blade must be changed, resulting in uneven machining allowance during finish grinding and it is difficult to ensure the required machining accuracy. In addition, the poor linearity (about 0.015mm) of the diamond grinding wheel used for precision grinding roller razors is also one of the important causes of tool manufacturing errors.
In summary, the tooth profile error caused by the tool manufacturing error (measurement error value of about 0.015 to 0.020 mm) is: Df tool = 0.015 to 0.020 mm.
3 Tooth shape error due to machine tool motion error
In hard tooth gear machining, the movement error of the processing machine has a great influence on the tooth profile error of the workpiece. Through inspection of the tested workpieces, it was found that the tooth shape error of the hard tooth face gear machining fluctuates compared with the ordinary gear processing, indicating that the hard tooth surface hobbing precision is very sensitive to the gear hardness, and the dynamic characteristics of the processing system are Tooth error has a decisive influence. In the motion error of the machine tool, the most significant influence on the tooth profile error is the torsional vibration error of the machine tool table and the displacement error of the front and rear of the machine tool in the radial direction of the gear.
1. Tooth shape error due to torsional vibration of the machine table
The torsional vibration of the gear processing machine table will directly affect the tooth profile error of the machined gear. This error is detected in the tangential direction of the gear. The calculation figure is shown in Figure 5. In the drawing, the workpiece is the diameter of the machined gear, Da< is the torsional vibration angle of the workpiece when the machine tool is cutting, and Da is the torsional vibration angle of the workpiece when the machine tool is not being cut.
From Fig. 5, it can be seen that the transmission error when the machine tool is not cutting is DS1'=r workpiece tanDa'=(176.25/2)tan0.0083°=0.013mm; the movement error when the machine tool is cutting is DS1=r workpiece tanDa=(176.25/ 2) tan0.021°=0.032mm. In the formula, Da'=0.0083° and Da=0.021° are measured experimentally. Since DS1
Through the test, the displacement error of the front and rear columns of the machine tool in the radial direction of the gear is DS2=0.015mm (see Figure 6). The resulting tooth profile error Dfs=DS2cos20°=0.014mm.
Therefore, the tooth profile error caused by the machine motion error is
Df machine tool = [(Dfs)2+(DS1)2]1â„2=[(0.014)2+(0.032)2]1â„2=0.035mm
The tooth profile error caused by tool manufacturing error and machine motion error is
Dff=[(Df tool)2+(Df machine tool)2]1â„2=(0.0202+0.0352)1â„2=0.041mm
Fig. 6 Deformation of the front and rear pillars of the machine tool in the radial direction of the gear
Figure 7 blade positioning slot structure
Figure 8 The tooth profile
Figure 9 Torsion angle of gear mounting shaft under cutting force
4 Measures to Reduce Hardened Tooth Profile Errors
1. Reduce tool manufacturing errors
In order to reduce the manufacturing error of gear cutting caused by gear cutting tools, it is necessary to improve the manufacturing precision of the hard tooth rolling razor. To this end, the following measures can be taken: 1 To improve the precision of the preform of the blade blank; 2 To change the blade to a positioning groove Structure (see Figure 7); 3 Calculate the knife edge curve, and open the front and back knife faces of the knife edge according to the approximate curve.
2. Reduce machine movement error
1. Hard-tooth hobbing process judgment In the hard-tooth hobbing process, if the machine tool movement error is too large, it will seriously affect the normal cutting. Generally, it can be judged whether the cutting is normal or not by observing the tooth profile. If the knife pattern is neat and continuous (see Fig. 8a), it indicates that the cutting is normal; if the knife pattern is hidden, there are traces of squeezing (see Fig. 8b), which indicates that some of the cutting edges are not normal; if the tooth surface is disordered or not visible at all Knife marks indicate that the cutting is abnormal.
2. Calculation of Machine Motion System Error
The dynamic error in the hard-tooth hobbing process is mainly manifested in the torsional vibration of the machine tool and the corresponding deformation of the support member. When cutting gears with carbide tools, the range of feed of machining teeth is usually f = 0.1-0.2mm/r, and the corresponding cutting force range is 15-22kg. For hardened gears with hardness greater than 60HRC, the cutting force corresponding to f = 0.1mm/r is 150kg; the cutting force corresponding to f = 0.2mm/r can reach 200kg. Generally, the hard-tooth roll cutting feed rate is f=0.2-0.4mm/r, so the amount of torsional vibration of the computer bed and the deformation of the supporting member caused by the cutting force of 200kg can be used.
1. Torsional vibration due to cutting force
As shown in Fig.9, the gear mounting shaft will produce a torsion angle Y under the action of cutting force F (F=200kg). Its calculation formula is
Ymax = (Mnmax/GIp) (180°/p)
Where: Mnmax - the maximum torque applied to the gear mounting shaft, Mnmax = Fd/2 = 1.725 × 105 N · mm
G——Shear elastic modulus, taking G=79.4 GPa
Ip - the moment of inertia of the gear mounting shaft, Ip=pd04/32=1.27×106mm4
d - gear diameter, d = 176.25mm
Figure 10 Force model of gear mounting shaft
D0——diameter of gear mounting shaft, d0=60mm
With the substitution parameters, Ymax=0.98°/m can be obtained, and the twist angle Y′max=0.15° at 150 mm of the gear mounting shaft. From this, it is possible to calculate the amount of torsional vibration of the gear mounting shaft caused by the cutting force DS=d/2tanY'max=176.25/2 tan 0.15°=0.23 mm. From the calculation results, it can be seen that under the action of the cutting force F, the torsion angle (Y'max=0.15°) of the gear mounting axis (Ø60 mm) is large, which causes a large error in the machine motion system. In order to solve this problem, a Ø120mm sleeve can be installed on the gear mounting shaft. The torsion angle Y'max=0.0098° obtained by this calculation can greatly reduce the error of the machine motion system.
2. Bending deformation of gear mounting shaft caused by cutting force
The cutting force F is converted into a force P acting in the radial direction of the gear, P=F/tanβ, and the tooth angle β=20°, then P=200/tan20°=550 kg. When calculating the radial displacement of the gear under the action of P=550kg, the stress condition of the gear mounting shaft is simplified to the model shown in Fig. 10. Based on this, the bending deformation of the gear mounting shaft can be calculated as
Yc=Yc1-Yc2
Where: Yc1 - Deflection due to force P at point C, Yc1 = Pa3/3EI
Yc2——deflection caused by PB at C point, Yc2=PBa2(3L-a)/6EI
I - section moment of inertia of the shaft, I=pd4/64
E——elastic modulus of the shaft, E=206GPa
L - the total length of the shaft, L = 660mm
A—distance of cutting force point C to fixed end A, a=380mm
D——The diameter of the shaft, D=60mm
PB——Support force at point B, PB=Pa2(3L-a)/2L3=221kg
1
Figure 11 Rear column force model
By substituting each parameter value into the formula, the bending distortion of the gear mounting shaft caused by the cutting force can be calculated as Yc=0.117mm.
3. Post-bending deformation caused by cutting force
The bending deformation of the rear column caused by the cutting force can be calculated according to the force model shown in Figure 11.
YB=PBa3/3EI
In the formula: PB—B point reaction force, PB=221kg
a—The distance between the fixed end and the support hole, a=660mm
E—elastic modulus of the column, E=135 GPa
I—section moment of the column, I=( BH3-bh3)/12=9.7x108mm4
B, H - column section size, B = 550mm, H = 335mm
t——The wall thickness of the column, t=32mm
b=B-2t=550-2×32=486mm,
h=H-2t=335-2×32=271mm
Substituting each parameter value into the formula, the post-pillar bending deformation amount YB=0.0016mm caused by the cutting force can be calculated.
From the above analysis and calculation, it can be seen that the gear mounting shaft and the column stiffness of the processing machine have a great influence on the processing accuracy of the gear. Therefore, in order to effectively reduce the tooth profile error of the gear being machined, it is necessary to increase the rigidity of the gear mounting shaft and the column, and to avoid or reduce the torsional vibration and the position change of the support member under the action of the cutting force as much as possible.
After the semi-finishing of the hardened gear tooth surface with a hard alloy roller razor, a fine grinding process is also required to ensure the gear machining accuracy. Through the precise installation and commissioning of the process system (machine tools—fixtures—tools—workpieces), the machining requirements for hardened tooth surfaces can be guaranteed.
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